Rigid resolution of a finitely generated module over a regular local ring (Q553427)

Let \((R, \mathfrak n)\) be a regular local ring and \(M\) a finitely generated \(R\)-module with \(\mathfrak n\)-stable filtration \(\mathbb M = \{M_i\}_{i \geq 0}\). We can construct a free resolution of \(M\) over \(R\) from the minimal free resolution of the associated graded module \(\text{gr}_{\mathbb M}(M)\) over the polynomial ring \(P = \text{gr}_{\mathfrak n}(R)\). In particular, the \(i\)th Betti number \(\beta_i(M)\) of \(M\) over \(R\) is at most one \(\beta_i(\text{gr}_{\mathbb M}(M))\) of \(\text{gr}_{\mathbb M}(M)\) over \(P\). In the present paper, authors give a sufficient condition for \(\beta_i(M) = \beta_i(\text{gr}_{\mathbb M}(M))\). That is, if, for some \(j\), \(\beta_j(M) = \beta_j(\text{gr}_{\mathbb M}(M))\) and the submodule of \(\text{gr}_{\mathbb M}(M)\) generated by the \(n\)th homogeneous component has a linear resolution for any \(n\), then \(\beta_i(M) = \beta_i(\text{gr}_{\mathbb M}(M))\) for every \(i \geq j\).